Managing Corporate Liquidity:
Strategies and Pricing Implications
Attakrit Asvanunt Mark Broadie Suresh Sundaresan ∗
June 29, 2010
Abstract
Defaults arising from illiquidity can lead to private workouts, formal bankruptcy pro-ceedings or even liquidation. All these outcomes can result in deadweight losses. Cor-porate illiquidity in the presence of realistic capital market frictions can be managedby a) equity dilution, b) carrying positive cash balances, or c) entering into loan com-mitments with a syndicate of lenders. An efficient way to manage illiquidity is to relyon mechanisms that transfer cash from “good states” into “bad states” (i.e., financialdistress) without wasting liquidity in the process. In this paper, we first investigate theimpact of costly equity dilution as a method to deal with illiquidity, and characterizeits effects on corporate debt prices and optimal capital structure. We show that equitydilution is in general inefficient. Next, we consider two alternative mechanisms: cashbalances and loan commitments. Abstracting from future investment opportunities andshare re-purchases, which are strong reasons for corporate cash holdings, we show thatcarrying positive cash balances for managing illiquidity is in general inefficient relativeto entering into loan commitments, since cash balances a) may have agency costs, b)reduce the riskiness of the firm thereby lowering the option value to default, c) post-pone or reduce dividends in good states, and d) tend to inject liquidity in both goodand bad states. Loan commitments, on the other hand, a) reduce agency costs, andb) permit injection of liquidity in bad states as and when needed. Then, we studythe trade-offs between these alternative approaches to managing corporate illiquidity.We show that loan commitments can lead to an improvement in overall welfare andreduction in spreads on existing debt for a broad range of parameter values. We deriveexplicit pricing formulas for debt and equity prices. In addition, we characterize theoptimal draw down strategy for loan commitments, and study its impact on optimalcapital structure.
∗Asvanunt, [email protected], Department of Industrial Engineering and Operations Re-search, Columbia University, New York, NY 10027; Broadie, [email protected], and Sundaresan,[email protected], Graduate School of Business, Columbia University, New York, NY 10027. This workwas supported in part by NSF grant DMS-0914539 and a grant from the Moody’s Foundation.
1
1 Introduction
Theory of optimal capital structure and debt valuation has made rapid strides in recent
years. Beginning with the pioneering work of Merton (1974), which provided the first sat-
isfactory formulation of debt pricing, a number of recent papers have extended the analysis
to consider the valuation of multiple issues of debt, optimal capital structure, debt rene-
gotiation, and the effects of Chapter 7 and Chapter 11 provisions on debt valuation and
optimal capital structure.1 These models have tended to abstract from cash holdings by
companies or loan commitments held by the companies. Yet these are important ways in
which companies deal with illiquidity and avoid potentially costly liquidations or financial
distress. In the banking literature, researchers have examined the role of loan commitments
and their effects on the welfare of corporate borrowers both in theoretical and empirical
work. Papers by Maksimovic (1990), Shockley (1995), Shockley and Thakor (1997), Sufi
(2007a) and Sufi (2007b) have explored the important role played by loan commitments
under capital market frictions. The insights of these papers have not yet been brought to
bear on the pricing and optimal capital structure implications. This is one of the goals of
our paper. A typical company often issues multiple types of debt, with different seniority
and covenants. As a result, cash flows to creditors are complicated to model when the firm
enters bad states (as in financial distress). In the case of default, the U.S. Bankruptcy
Codes (Chapter 11, 13, etc.) allow companies to postpone liquidation by providing the
borrowing firm with protection while it undergoes reorganization and debt renegotiation
with the creditors. Alternatively, companies can also enter into financial contracts such as
loan commitments in good states (when they are solvent), to alleviate their financial dis-
tress and thus postponing or avoiding bankruptcy. In this paper, we propose a structural
model that incorporates loan commitments as an alternative source of liquidity. We con-
trast this strategy with two other mechanisms that have been studied in the literature for
managing illiquidity: cash balances and costly equity dilution. Our paper does not consider
dynamic investment opportunities (future growth options), and this assumption precludes1An important subset of this literature includes the papers by Black and Cox (1976), Leland (1994),
Anderson and Sundaresan (1996), and Broadie et al. (2007).
2
an important incentive for holding cash: the model presented is more relevant for mature
firms.
1.1 Overview of Major Results
In our paper we integrate the insights of the banking literature on loan commitments with
the insights of debt pricing literature and provide a framework for analyzing loan com-
mitments as a part of optimal capital structure. Our paper also formally explores three
distinct ways in which companies deal with illiquidity – equity dilution, cash balances and
loan commitments. To our knowledge, this is the first paper to undertake this compre-
hensive examination of corporate strategies for managing liquidity. In so doing, we extend
the theory of optimal capital structure to permit junior and senior debt, where the cor-
porate borrower can optimally select the “draw down” and “repayment” strategy of loan
commitments.
We provide an analytical characterization of credit spreads, optimal capital structure and
debt capacity when equity dilution is costly, subject to the endogenous default boundary
as a solution to a nonlinear equation. Using the base case of zero cost of equity dilution,
we characterize the welfare losses due to dilution costs when equity holders decide on the
choice of the optimal default boundary.
We show, through an induction argument, that it is never optimal to carry cash balances to
guard against default when equity issuance is costless. More importantly, we characterize
the threshold levels of managerial frictions associated with holding cash and equity dilution
costs over which it is never optimal to hold cash balances. Our results show that the dilution
costs of equity have to be very significant and the costs of holding cash have to be very low
in order for the firm to curtail dividends and hold cash balances.
In the context of previous results, we explore contingent transfer of liquidity through the
use of loan commitments where the borrower is able to pay a commitment fee in good
states and draw down cash in bad states as needed. We develop a model and computa-
tional methodology needed for loan commitments. Using this approach, we characterize
3
the optimal draw down and repayment strategy and the effects of loan commitments on
existing debt spreads and the welfare of the borrowing firm. We find that entering into a
loan commitment agreement can benefit both equity and debt holders. We also study the
impact of a loan commitment on the firm’s optimal capital structure.
Our results should be viewed in the context of our model assumptions. We treat the firm’s
investment policy as fixed and exogenous. Availability of future investment and growth
options may provide a strong motive for accumulating cash balances when equity dilution
costs are significant. We abstract from this important consideration. Our paper does
not model strategic debt service considerations. Since strategic debt service (especially
when renegotiations are not costly) allows for underperformance in bad states, it acts as a
substitute to holding cash balances. This has been noted by Acharya et al. (2006). Our
analysis, however, can be modified to incorporate strategic debt service. Finally, we do not
model the Chapter 11 provision of the bankruptcy code, which also allows illiquid firms to
seek bankruptcy protection and coordinate their liquidity problems with lenders and avoid
potential liquidation.
1.2 Review of The Literature
The structural approach to modeling corporate debt begins with the seminal work of Merton
(1974). He indicates that risky zero-coupon bonds may be valued using the Black-Scholes-
Merton option pricing approach. In this model, default occurs if the asset value of the firm
falls below the debt principal at maturity. Black and Cox (1976) extend this model for
coupon bonds with finite maturity, where they impose that the firm must issue equity to
make the coupon payments, and that default is endogenously chosen by the firm at any
time prior to maturity. Leland (1994) derives a closed-form solution for prices of perpetual
coupon bonds. His model incorporates bankruptcy cost, tax benefit of coupon payments,
and dividend payouts. He also derives the expressions for optimal capital structure, debt
capacity, and optimal default boundary. Broadie et al. (2007) consider the presence of Chap-
ter 7 and Chapter 11, which leads to reorganization upon bankruptcy prior to liquidation.
All of these models restrict the firm from selling its asset.
4
The work mentioned above implicitly assume that liquidity has no value. This follows
from the assumption that the firm can dilute equity freely to finance short-fall of cash
from operations to fund the coupon payments. Equity issuance, however, is quite costly in
practice, and hence many firms maintain cash reserves or enter into agreements such as a
loan commitment to manage liquidity crises.
The literature of cash reserves is related to that of dividend policies. Since the work of
Miller and Modigliani (1961), who show that dividend policy is irrelevant to the firm value
in perfect (frictionless) markets, extensive research has emerged in the study of optimal div-
idend policies by means of optimization and stochastic optimal control under more realistic
market conditions. Examples of stochastic control models include Asmussen and Taksar
(1997), Jeanblanc-Picque and Shiryaev (1995), and Radner and Shepp (1996). These mod-
els choose a policy that maximizes the present value of a future dividend stream subject
to various cost structures associated with the distribution. Taksar (2000) provides a more
detailed discussion of the literature in this area. Cadenillas et al. (2007) model the cash
process as a mean-reverting process, and characterize optimal dividend policy in terms of
the amount and time of dividend distributions. Our model is more similar in spirit to those
of Kim et al. (1998) and Acharya et al. (2006), where the focus is on liquidity. Kim et al.
(1998) develop a three-period model that investigates the trade-offs between holding liquid
assets (cash) with lower returns versus facing future uncertainty and costly external fund-
ing (equity issuance). They find that the optimal level of cash is increasing with the cost
of equity issuance and the volatility of future cash flows. Acharya et al. (2006) develop a
multi-period model that allows the firm to simultaneously keep cash reserves and service
debt strategically. They find that the option to carry cash to is valuable if the firm also
has the option for strategic debt-service. We provide a fully dynamic model of endogenous
default and derive implications for cash balances and line of credit.
Existing literature on loan commitments dates back to Hawkins (1982) and Ho and Saunders
(1983). Hawkins (1982) develops a pricing framework for loan commitment contracts under
two different models. The first model assumes that the firm’s assets are bought or sold
when the loan commitment is drawn down or repaid, respectively. The second assumes
5
that the shareholders receive dividends when loan commitment is drawn down, and new
shares are issued when it is repaid. Ho and Saunders (1983) focus on the hedging of a loan
commitment with bond futures contracts. In their model, they assume that the firm has
an option to draw down at a fixed time, and cannot repay it before maturity. Pedersen
(2004) discusses these two papers in more detail. Martin and Santomero (1997) formulate
an optimization problem for a firm which enters into a loan commitment for future financing
of investment opportunities that arrive stochastically. They discuss the optimal size of the
loan commitment, the determinants of the size, and the average usage by the firm. Pedersen
(2004) investigates the impact of loan commitment on existing debt. He imposes a covenant
that restricts dividend payouts to shareholders when the loan commitment is drawn. In this
model, excess cash from revenue after making coupon payments must go back to repay the
loan commitment immediately. Our model differs from the above, with the exception of
Pedersen (2004), in terms of the purpose of the loan commitment. We use loan commitment
solely for managing liquidity, and not for financing investment opportunities. In addition,
our model accounts for both the optimal draw down and repayment strategies of the loan
commitment.
There is also a body of literature that discusses the roles and benefits of loan commitments.
Maksimovic (1990) argues that loan commitments can increase individual firm’s profit in an
imperfectly competitive market, when they are used to finance the firm’s strategic position
as a response to its rival’s decisions. Berkovitch and Greenbaum (1991) develop a two-
period model to show that a loan commitment may be an optimal way to finance risky
investments. Shockley (1995) demonstrates that a loan commitment leads to higher optimal
debt level and lower cost of debt under similar settings. Snyder (1998) argues that a loan
commitment is a strictly better source of financing than a standard debt, in the presence
of the debt overhang problem originally proposed by Myers (1977), because of its fixed-fee
and low interest rate features. Under our model assumptions, we also find that the loan
commitment increases the optimal level of existing debt and may reduce its spread.
On the empirical side, Melnik and Plaut (1986) provide empirical analysis on “packages” of
loan commitment terms from surveyed data of 132 U.S. non-financial firms. They find that
6
the loan commitment size is positively correlated with the interest mark-up, commitment
fee, length of contracts, current ratios, and firm size. Shockley and Thakor (1997) provide
similar analysis based on data of 2,513 bank loan commitment contracts sold to publicly
traded U.S. firms. The data were collected from Securities and Exchange Commission
filings by the Loan Pricing Corporation. Agarwal et al. (2004) test the model of Martin and
Santomero (1997) empirically against the data for loan commitments made to 712 privately-
owned firms, and find the results consistent with the model’s predictions. Sufi (2007a) has
presented evidence that suggests that banks are unwilling to provide loan commitments to
firms that do not have sufficient operating cash flows. In other words, there is a supply side
constraint on the ability of borrowing firms to get loan commitment.
Loan commitments often carry Material Adverse Change (MAC) provisions, which may
allow the bank to withdraw the lines under specified economic circumstances. This said,
MAC is rarely invoked: firms such as Enron were able to draw their lines before they
eventually declared bankruptcy.2 James (2009) notes that the drawdown of credit lines may
help explain the fact that bank commercial and industrial (C&I) lending increased during
the fall of 2008 when borrowers and bankers were reporting extremely tight credit market
conditions. C&I loans held by U.S. domestic banks increased from $1.514 trillion in August
2008 to $1.582 trillion by year-end, while the Federal Reserves Senior Loan Officer Opinion
Survey on Bank Lending Practices found that a record net 83.2% of respondents reported
that their institutions had tightened loan standards for large and medium borrowers in the
fourth quarter. This pattern of usage of loan commitments is consistent with the theory
developed in our paper.
The remainder of the paper is organized as follows. Section 2 describes the basic setup of
the model. In section 3, we study the welfare and pricing of the firm’s securities under
costly equity dilution. We introduce the loan commitment into the model in in Section 4.
Section 5 concludes and the Appendices contain technical proofs and our computational
methodology.2See James (2009), for example.
7
2 The Model
In a perfect capital market where there are no frictions and where distress can be resolved
costlessly, the question of corporate liquidity is not relevant. Empirical evidence suggests,
however, that there are at least two important sources of frictions which render corporate
liquidity a very important question. First, there are costs associated with security issuance.
Such costs reflect not only the costs associated with underwriting and distribution, but also
with asymmetric information that characterize the relationship between the suppliers of
capital and corporations. A survey by Lee et al. (1996) finds that direct equity issuance
cost (underwriter spreads and administrative fees) on average is over 13% for $2-10 million
and over 8% for $10-20 million of seasoned equity offerings. The average cost for initial
public offerings is even higher, 17% for $2-10 million and over 11% for $10-20 million. In
addition, financial distress can lead to private workouts, bankruptcy proceedings and even
liquidation, all of which are known to be costly. Branch (2002) reports total bankruptcy-
related costs to firm and claimholders to be between 13% and 21% of pre-distressed value.
Bris et al. (2006) provide a detailed analysis of bankruptcy costs under Chapter 7 and 11.
They find that the average change in asset value maybe as high as 84% before and after the
filing of Chapter 7. In this paper, we construct a simple trade-off model of capital structure,
where corporate liquidity plays a pivotal role. Our structural model is an extension of Leland
(1994) with net cash payouts by the firm (Section VI. B). We introduce two costs explicitly.
First, we model the cost of maintaining liquidity by introducing an exogenous cost on cash
balances. In addition, we also recognize that equity dilution is costly. A corporate borrower
chooses a level of debt by trading off tax benefits and the possibility of costly liquidation
in the future. We ask how this decision is informed by managing the liquidity of the firm
through three possible tools: a) equity dilution, b) carrying cash balances, and c) using loan
commitment, which are fairly priced. In order to focus squarely on liquidity strategies of the
corporation in a trade-off model, we abstract from a detailed modeling of the bankruptcy
code, which in itself is an institution that is designed, in part, to address the question of
resolving illiquidity. It will be of interest to extend our analysis to examine how the liquidity
8
strategies of a firm will be influenced by the availability of a well designed bankruptcy code.3
Since the investment policy is fixed, we restrict the sale and purchase of the firm’s assets,
and hence the firm can only use its revenue or proceeds from equity dilution to make the
coupon payments in absence of cash balances or loan commitments. We consider explicit
costs associated with holding cash explicitly. As an alternative to cash or equity dilution,
the firm may also enter into a loan commitment to transfer cash from the solvent state to
the state when the firm is not generating enough revenue to cover the coupon payment. We
assume that the firm has to pay a one-time fee at the time the loan commitment contract
is signed. Since the firm’s asset cannot be purchased or sold, the sole purpose of cash and a
loan commitment is to provide liquidity when the firm is in distress, and not for investment
purposes.
2.1 Basic Setup
We assume that the firm’s asset value, denoted by Vt, is independent of its capital structure,
and follows a diffusion process whose evolution under the risk neutral measure Q is given
by:
dVt = (r − q)Vtdt+ σVtdWt
where r is the risk-free rate, q is the instantaneous rate of the return on asset, σ is the
volatility of asset value, and Wt is a standard Brownian motion under Q. The instantaneous
revenue generated by the firm is given by:
δt = qVt.
The parameter q determines the internal liquidity of the firm from its cash flow. Holding
the current asset value fixed, a higher value of q implies that the firm produces more cash
flow per unit time than a lower value of q. In this regard, we may interpret a firm with a
small q as being a “growth” firm, and a firm with a large q as being a “mature” firm.3Examples of papers that examine alternative bankruptcy procedures include Aghion et al. (1992), Be-
bchuk (1988), Bebchuk and Fried (1996) and Hart et al. (1997). Hart (2000) summarizes the generalconsensus on the goals and the characteristics of efficient bankruptcy procedures.
9
The firm issues debt with principal P , a continuous coupon rate c, and a maturity T . The
firm uses its revenue δt to make the coupon payment, and uses any surplus to pay dividends
to the shareholders. In absence of cash balances and loan commitment, when revenue is
not enough to cover the payment, the firm can either dilute equity to raise money to make
the payment or declare bankruptcy. In the latter case, the firm liquidates its assets and the
proceeds go to the debt holders. There is an equity dilution cost of γ and a liquidation cost
of α, which will be explained in the next section. We next describe the three strategies to
manage corporate illiquidity.
2.2 Equity Dilution
When faced with a liquidity crisis, the firm may raise capital by issuing additional equity.
We assume that the cost of equity dilution is proportional to the proceeds from the issuance.
Let γ be the proportion of the proceeds lost due to deadweight costs of issuance. In other
words, if x is the percentage of outstanding shares to be issued, then the proceeds from
equity dilution are x1+xE. However, the firm only receives (1− γ) x
1+xE. The claim value of
the original shareholders reduces to 11+xE.
Suppose the firm needs to raise C dollars from equity dilution, then the percentage of shares
it needs to issue is given by:
x =C
(1− γ)E − C
and the claim value of the original shareholders reduces to:
11 + x
E =(
11 + (C/((1− γ)E − C))
)E = E − C
1− γ
which is equivalent to the shareholders receiving a negative dividend of C1−γ . In our model,
equity issuance would not be undertaken if it does not lead to an increase in the value of
equity.
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2.3 Cash Balance
An alternative to issuing equity is to build up a cash balance. The firm may choose a
dividend payout rate d strictly less than the revenue rate δ in the good states when the
revenues exceed coupon payments, and thus accumulate a cash balance, x, over time. On
this cash balance, we assume that the firm earns an interest at rate rx ≤ r, reflecting the
possible costs associated with leaving cash inside the firm. In our model, the firm chooses
the payout rate d that maximizes its equity value.
2.4 Loan Commitment
Finally, the firm may enter into a loan commitment with another lender. A loan commitment
is a contractual agreement between lender (the “bank”) and the firm, that gives the right,
but not an obligation, to the firm to borrow in the future at the pre-specified terms. The
terms may include, but are not limited to, the size of a loan commitment P l, a rate of
interest (coupon rate) cl, and an expiration date. Generally, there are drawn and undrawn
fees associated with the loan commitment.
In our model, the bank imposes a covenant that forbids the firm from using the proceeds of
the loan to pay shareholder dividends. The firm does not keep a cash balance, so as a result,
it only considers drawing down the loan when revenue is not sufficient to cover the coupon
payment. Furthermore, the firm can at most draw down the amount of the shortfall. The
bank also allows the firm to repay any amount of the loan at any time prior to maturity.
We assume that the firm makes all its decisions to maximize the equity value.
We describe in detail how the firm makes the decisions, and how the fees are determined in
Section 5.
3 Welfare and Pricing Under Costly Equity Dilution
In this section, we show how equity, debt and total firm value can be determined when
equity dilution is costly. We derive closed-form solutions for these values for the infinite
11
maturity case. We show how equity dilution cost affects credit spread, debt capacity and
optimal capital structure.
3.1 Exogenous Bankruptcy Boundary
First, we derive closed-form solutions for equity, debt and total firm values when the
bankruptcy boundary is determined exogenously. We assume that there is an asset value
boundary VB, below which the firm will declare bankruptcy. Upon bankruptcy, the firm
liquidates its assets and incurs a liquidation cost of α, i.e., the proceeds from liquidation
are (1− α)(V + δ).
It is well known (e.g. Duffie (1988)) that the value, f(Vt, t), of any security whose payoff
depends on the asset value, Vt, must satisfy the following PDE:
12σ2V 2
t fV V + (r − q)VtfV − rf + ft + g(Vt, t) = 0 (1)
where g(Vt, t) is the payout received by holders of security f .
In the perpetuity case, the security becomes time-independent, and hence the above PDE
reduces to the following ODE:
12σ2V 2fV V + (r − q)V fV − rf + g(V ) = 0 (2)
Equation (2) has the general solution:
f(V ) = A0 +A1V +A2V−Y +A3V
−X (3)
where
X =
(r − q − σ2
2
)+
√(r − q − σ2
2
)2+ 2σ2r
σ2(4)
Y =
(r − q − σ2
2
)−√(
r − q − σ2
2
)2+ 2σ2r
σ2(5)
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3.1.1 Equity Value
Consider a firm that issues perpetual debt which pays a continuous coupon at the rate
C. When the firm’s revenue rate exceeds the coupon rate, qV ≥ C, equity holders receive
dividends at rate (qV −C), otherwise the firm dilutes equity to make the coupon payment.
As explained above, the latter is equivalent to a negative dividend of β(qV − C), where
β = 1/(1− γ).
Therefore, the equity value, E, must satisfy:
12σ2V 2EV V + (r − q)V EV − rE + β(qV − C) = 0 for VB ≤ V < C/q (6)
12σ2V 2EV V + (r − q)V EV − rE + qV − C = 0 for V ≥ C/q (7)
The general solution (3) becomes:
E(V ) =
A0 +A1V +A2V−Y +A3V
−X for VB ≤ V < C/q
A0 + A1V + A2V−Y + A3V
−X for V ≥ C/q(8)
To solve for the coefficients in (8), we use the fact that E(V ) must also satisfy the following
boundary, continuity and smooth pasting conditions:
(BC I): E(VB) = 0
(BC II): limV→∞E(V ) = V − Cr
(CC): E((C/q)−) = E(C/q)
(SP): EV ((C/q)−) = EV (C/q)
(9)
(BC I) follows from our assumption that the shareholders get nothing when the firm declares
bankruptcy. (BC II) is the value of equity when the firm becomes riskless as the asset value
approaches infinity. (CC) and (SP) ensures a smooth and continuous transition in and out
of equity dilution region. This follows from the fact that the equity value fully anticipates
the future events.
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By solving the system of equations above, the equity value is given by:
E(V ) =
β(V − C
r
)+A2V
−Y +A3V−X for VB ≤ V < C/q(
V − Cr
)+ A3V
−X for V ≥ C/q(10)
where
A2 =(β − 1)
(Cq
)Y (X C
r −XCq −
Cq
)X − Y
A3 =[β
(C
r− VB
)−A2V
−YB
]V XB
A3 =
[(β − 1)
(C
q− C
r
)V −XB + β
(C
q
)−X (Cr− VB
)]V X+YB
+(β − 1)
(X C
r −XCq −
Cq
)(V −XB − V −YB
(Cq
)Y−X)X − Y
V X+YB
The details of this derivation can be found in Appendix A.
3.1.2 Debt Value
Debt holders receive a constant payout rate of C as long as the firm remains solvent, i.e.,
when V ≥ VB. Therefore the debt value, D, must satisfy:
12σ2V 2DV V + (r − q)V DV − rE + C = 0 for V ≥ VB (11)
Equation (11) has a general solution:
D(V ) = B0 +B1V +B2V−X (12)
where X is given by (4).
Similar to solving for the equity value, we use the fact that D(V ) must satisfy the following
14
boundary conditions:
(BC I): D(VB) = (1− α)VB
(BC II): limV→∞D(V ) = Cr
(13)
(BC I) follows from our assumption on bankruptcy cost, and (BC II) is the value of a
perpetual debt as it becomes riskless.
Then we can show that the debt value is given by:
D(V ) =C
r+(
(1− α)VB −C
r
)(V
VB
)−Xfor V ≥ VB (14)
The details of this derivation can be found in Appendix A.
3.2 Endogenous Bankruptcy Boundary
In the previous section, we assume that the bankruptcy boundary is determined exogenously.
In practice, the management of the firm acts in the best interest of its shareholders. Thus
in the absence of a covenant imposed by the debt holders, the firm will choose a bankruptcy
boundary that maximizes its equity value. From the expression of equity value given in (10),
we can see that the equity value only depends on VB through A3 (assuming that optimal
VB is in the equity dilution region, V < Cq ). Thus, the optimal VB can be found by solving
∂E∂VB
= 0, which is equivalent to solving ∂A3∂VB
= 0. Consequently, the optimal bankruptcy
boundary, V ∗B, is given by the following implicit expression:
XβC
r(V ∗B)−1 − (1 +X)β − (X − Y )A2(V ∗B)−Y−1 = 0 (15)
The equity and debt values with endogenous bankruptcy boundary is therefore given by (8)
and (14), evaluated at V ∗B.
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3.3 Impacts of Costly Equity Dilution
3.3.1 Security Values
Figure 1 plots equity and debt values as a function of the underlying asset value as given
by (8) and (14). We observe that the optimal bankruptcy boundary, VB, increases as γ
increases. This makes sense because as asset value declines, the equity value declines at a
faster rate with costly equity diltuion, and hence bankruptcy is declared at a higher asset
value. Equity value is also decreasing in γ. The impact of dilution cost converges to zero
as V →∞.
We observe that debt value is decreasing in γ. Since D(V ) is only a function of γ through
VB and VB is increasing in γ, from (14), we can deduce that D(V ) is decreasing in γ if and
only if (1−α)VB ≤ Cr
X1+X . Similar to the equity value, the effect of dilution cost converges
to zero as V →∞.
Figure 1: Plots of equity, E(V ), and debt values, D(V ), as a function asset value at different levels equitydilution cost, γ. Both E(V ) and D(V ) are increasing in V , decreasing in γ, and converge to the Lelandresults as V →∞ for all values of γ. (V = 100, C = 3.36, σ = 0.2, r = 0.05, q = 0.03, α = 0.3, τ = 0.15)
3.3.2 Debt Capacity and Optimal Capital Structure
The debt capacity of a firm is defined as the maximum value of debt that can be sustained
by the firm. This is determined by the value of coupon, Cmax, which maximizes the debt
16
value, denoted by Dmax. In this section, we illustrate numerically how the cost of equity
dilution, γ, affects Dmax. The left panel of Figure 2 shows that Cmax, as well as Dmax,
decreases as equity dilution becomes more costly. In this example, a dilution cost of 10%
reduces the debt capacity from $86 to $84.
The optimal capital structure of the firm is defined as the coupon value, C∗, that maximizes
the total firm value, denoted by v∗. The right panel of Figure 2 illustrates how the optimal
coupon, C∗, as well as the firm value, v∗, decreases as equity dilution becomes more costly.
Therefore, neglecting the costliness of equity dilution can result in over-leveraging the firm.
In this example, a dilution cost of 10% reduces the optimal coupon from $3.3 to $3 (9%).
This translates to a 0.6% change in the firm value.
Figure 2: The left panel plots debt value vs. coupon rate. This illustrates how the debt capacity is affectedby the equity dilution cost. The debt capacity decreases as equity dilution becomes more costly. The rightpanel plots total firm value vs. coupon rate. This illustrates how the optimal capital structure is affectedby the dilution cost. The firm issues lower coupon when dilution cost is higher under the optimal structure.(V = 100, σ = 0.2, r = 0.05, q = 0.03, α = 0.3, τ = 0.15)
4 Cash Balances to Manage Illiquidity
In this section, we explore the optimality of building cash balances in good states in order
to confront illiquidity in bad states of the world. This strategy may make sense when equity
dilution costs are high and the costs of holding cash balances are low. We model the costs
17
of holding cash by assigning a rate of return rx on cash balances, where rx ≤ r, and r is the
risk-free rate. We refer to the difference, rx − r, as the cost associated with holding cash
inside the firm. We continue to model equity dilution costs as in the previous section. Note
that the strategy of building cash balances has several disadvantages. First, building cash
balances requires that the equity holders forego some dividends in good states. Second,
the firm wastes liquidity by carrying cash in good states when it is not needed. Third, the
accumulation of cash balances makes the existing debt less risky and thereby may induce a
transfer of wealth from the equity holders to the existing creditors. We prove the following
general result:
Proposition 1. If there is no equity dilution cost and the interest on cash reserve is not
greater than the risk-free rate, then paying out the entire cash reserve as dividend is an
optimal strategy when the firm wishes to maximize its equity value.
The proof of this proposition is in Appendix B.4
This base case result establishes a benchmark for evaluating the following question: how
high must the equity dilution costs be in order for the firm to hold cash? Note here that
there are no other motives such as growth options or investments for carrying cash in
our setting. The results presented in the remainder of this section are obtained using the
binomial approach described in Appendix C. We use the infinite horizon implementation
with T = 200, N = 2,400 and M = 40.
Figure 3 shows the magnitude of equity dilution costs for various costs of carrying cash
under which it is optimal for the firm to carry cash balances. The indifference curve shows
that for high costs of equity dilution and low costs of holding cash inside the firm, cash
balances will be carried by the firm to avoid illiquidity induced liquidation. It also shows
that the region for holding cash is larger for the more volatile firm.4This result is similar to the Proposition 4 of Fan and Sundaresan (2000) where it is shown that it is
optimal to pay all extra cash flows in good states as dividends. But their result assumes that the cash flowsare reinvested in the technology of the firm and no cash balances were modeled. Similar result has beenshown by Kim et al. (1998) in a static setting.
18
Figure 3: This figure plots an indifference region for maintaining a positive cash balance for differentlevels of equity dilution and costs.of holding cash balances. The upper-left region is where it is optimal tocarry a positive cash balance, while the lower-right region is where it is optimal to always pay everythingout as dividend. The plot shows that the region for keeping cash is larger for firms with higher volatility.(V = 100, σ = 0.2, r = 0.05, q = 0.03, α = 0.3, τ = 0.15)
4.1 Optimal Level of Cash
In this section, we discuss the optimal level of cash for a firm that maximizes equity value.
Note that cash has the following pros and cons: carrying cash exposes the firm to costs and
forces equity holders to forgo dividends in good states. This makes the firm relatively safe,
benefiting the debt, ex-post. But, this latter effect also results in lower spreads, ex-ante,
which is good for the equity holders. The left panel of Figure 4 plots the equity values as a
function of the cash balance for different levels of costs of holding cash balances. It shows
that when there is no cost to holding cash balances, equity value is maximized as long as the
cash balance is above a certain level. However, with strictly positive agency cost, there is a
unique optimal level of cash. The right panel of Figure 4 plots the optimal level of cash as
a function of asset values for different levels of costs of holding cash balances. As expected,
the optimal level is decreasing with the asset value. We also observe that below a certain
level of asset value, the optimal cash level is zero. This is because the firm is anticipating
bankruptcy and is passing the cash to shareholders so that debt holders will not receive it
upon liquidation.
19
Furthermore, we observe that optimal level of cash decreases drastically when the cost of
holding cash balances increases from 0% to 1%. Later, we will see that the impact of cash
quickly diminishes as the cost increases for this reason.
Figure 4: The left panel plots equity value, E, vs. cash balance, x, for different levels of costs. With nocost, E is increasing in x and levels off beyond a certain value of x. With positive cost, E is increasing in x upto a critical value of x when it starts to decrease. This results in a unique optimal level of x that maximizesE. The right panel plots the optimal level of cash balance, x∗, vs. asset value, V . x∗ = 0 for values of Vbelow a certain critical level, and it is decreasing in V beyond that level. This critical level is close to thebankruptcy boundary. (V = 67, σ = 0.2, r = 0.05, q = 0.03, C = 3.36, α = 0.3, τ = 0.15, γ = 0.15)
4.2 Impacts of Cash Balance
In this section we discuss the effect of cash balances on debt and total firm value. The left
panel of Figure 5 shows that optimally carrying a cash balance increases debt value as well
as equity values. However, the impact becomes negligible when the cost is strictly positive.
When the cash balance is carried optimally, it can alter the optimal capital structure of
the firm as well. The right panel of Figure 5 illustrates this effect. The optimal capital
structure increases relative to the Leland benchmark (i.e., no cash balance or infinite cost
of holding cash). The availability of cash enables the firm to increase the proportion of debt
it holds in the optimal capital structure. Note that when the cost increases from zero to
1% there is a dramatic shift in the firm value as well as in the optimal level of coupon. For
subsequent increases in costs, there is only a marginal variation the firm values and optimal
20
coupon. This is because the optimal level of cash decreases rapidly when cost increases, as
illustrated in the right panel of Figure 4.
Figure 5: The left panel plots debt value vs. coupon rate. This illustrates how debt capacity is affectedby cash balance. Cash balance virtually has no effect on debt value when there is a positive cost of holdingcash. At zero cost, it increases the debt value. The right panel plots total firm value vs. coupon rate. Thisillustrates how the optimal capital structure is affected by the cash balance. Cash balance increases thevalue of the firm and also increases the optimal coupon rate. (V = 100, σ = 0.2, r = 0.05, q = 0.03, α =0.3, τ = 0.15, γ = 0.15)
We characterize the impact of optimal cash balances on spreads in Figure 6. We have seen
earlier that as costs of holding cash increase, the debt value decreases and the optimal
capital structure carries a lower proportion of debt. Note that the spreads decline as the
costs decline. This is due to the fact that higher cash balances can be optimally carried
at low costs. This makes the firm less risky and provides it with cushions from premature
liquidation, and the firm’s debt consequently becomes more attractive to investors.
Although the cash balance helps the firm avoid equity dilution, it does not significantly post-
pone the expected time to default, or its distribution, when carried optimally. This result
is somewhat surprising. But as we mentioned before, as the firm approaches bankruptcy, it
depletes the cash balance by paying shareholders dividend. The result is different when the
firm is maximizing total firm value instead of equity value. Appendix F discusses this dif-
ference in more detail. Also in Section 5.2, we will see that a loan commitment significantly
impacts the distribution of default times even in the equity maximizing case.
21
Figure 6: This figure shows how much cash balance reduces the spread of the existing debt. (V = 100, σ =0.2, r = 0.05, q = 0.03, α = 0.3, τ = 0.15, γ = 0.15)
We have characterized in this section how cash balances may be optimally used to manage
illiquidity in the presence of costs of holding cash balance and equity dilution costs. We
now turn to the use of a loan commitment as an alternative to managing illiquidity.
5 Loan Commitment
In this section, we show how the firm can use a loan commitment to manage its liquid-
ity problem and relieve it from expensive equity dilution costs and costs associated with
carrying cash.
We first define a stylized loan commitment contract. The firm pays a one-time fee (undrawn
fee), F , at the time it enters into the loan commitment. Let p be the amount of loan
commitment outstanding, and C l(p) be the associated coupon obligation (drawn fee). Then
when the firm faces a cash flow shortage, i.e., when its revenue is not sufficient to cover
the coupon payments, the firm has the option to draw down the loan commitment as long
as p < P l. We assume that the firm chooses between drawing down the loan and diluting
equity to maximize its equity value. The decision that the firm faces is how much to draw
down or repay the loan commitment at each time. We denote this decision variable by λ,
22
where λ > 0 corresponds to drawing down the loan and λ < 0 corresponds to repaying the
loan.
At any given time, if the firm is still solvent, then one of the following events can happen:
First, in the case where there is no cash flow shortage, i.e., δ ≥ C + C l(p), the firm can
decide how much of the outstanding loan commitment to repay, and the remaining cash
surplus goes to the shareholders as dividends. Note that since the firm can neither use the
proceeds from the loan for dividends, nor keep a cash balance, it cannot draw down the loan
commitment in this case. Hence λ must be in the interval [−p, 0]. If |λ| > δ − C + C l(p),
then the firm is also diluting equity to repay the loan commitment.
Second, in the case where there is a cash flow shortage, i.e., δ < C +C l(p), the firm has to
decide how to manage this shortfall. It needs to raise an additional C + C l(p)− δ to meet
its coupon obligations. The firm can decide between diluting equity or drawing down the
loan commitment, or both. Again, since we restrict the firm from using the proceeds from
the loan for dividends, it can draw down at most the shortfall amount, as long as it does
not exceed the credit line. We also restrict the firm from repaying the loan commitment in
this case. Thus λ must be in the interval [0,min{C + C l(p)− δ, P l − p}].
If the firm declares bankruptcy, then we apply an absolutely priority rule to the distribution
of proceeds from liquidation. The bankruptcy cost is shared between the bank and the debt
holders. Let L(V, p) be the value of the loan commitment. Then the value of the debt and
the loan commitment before liquidation costs are:
L(V, p) = min{V + δ, p}
D(V, p) = V + δ − L(V, p)
And then we apply the bankruptcy cost pro rata. Thus the debt and loan commitment
23
values ex-liquidation costs are:
L(V, p) = L(V, p)− L(V, p)L(V, p) + D(V, p)
α(V + δ)
D(V, p) = D(V, p)− D(V, p)L(V, p) + D(V, p)
α(V + δ)
The one-time fee, F , is chosen such that the loan commitment has zero value to the bank
at the time of signing. Therefore, it is the difference between the present value of the loan
commitment and the present value of all expected future draws on the loan commitment:
F =∫ T
0e−rtλ+
t dt− L(V0, 0) (16)
The coupon rate of the loan commitment, cl, is determined as follows. Given the current
capital structure of the firm, the bank first determines the level of firm’s asset value at
which it will start drawing down the loan (qV ≤ C). Then for a loan commitment of size
P l, the bank decides what the cost of borrowing for the firm should be in that state. In
particular, it chooses cl to solve
P l = D(C/q, clP l),
where D(V,C) is the value of a perpetual debt that pays continuous coupon of rate C dollars
issued by the firm with asset value V .
This choice of cl is reasonable because when the firm faces a liquidity crisis at V = C/q, its
alternative to a loan commitment is to issue additional perpetual debt. If the bank charges
any amount greater than cl, the firm might as well issue additional perpetual debt instead
of using a loan commitment.
The size of the loan commitment P l is usually chosen by the firm. Larger P l provides
more liquidity to the firm, but it also requires larger coupon C l and larger fee F . Thus the
firm chooses P l that maximizes the benefit to the shareholders. The left panel of Figure 7
illustrate this result for different levels of α, at the respective level of optimal coupon in
24
absence of the loan commitment. There is also a maximum value of P l that the firm can
support. This is similar to the concept of debt capacity. The right panel of Figure 7 shows
how the loan commitment capacity changes with liquidation cost, α.
The results in this section are obtained by using the binomial approach described in Ap-
pendix D. We use the infinite horizon implementation with T = 200, N = 2,400 and M =
40.
Figure 7: The left panel plots impact of loan commitment on equity value vs. size of the loan commitment,P l. The figure shows that there is an optimal value, P l∗, where shareholders benefit the most. Furthermore,P l∗ as well as the impact itself decreases as liquidation cost, α, increases. The right panel plots firm’s capacityfor loan commitment vs. liquidation cost. The firm is able to take on smaller size of loan commitment as αgets larger. (V = 100, C = 3.17, σ = 0.2, r = 0.05, q = 0.03, τ = 0.15, γ = 0.05)
5.1 Optimal Draw Down and Repayment Decisions
When a loan commitment is available to the firm, we find that it will gradually draw it down
until the credit limit is reached before diluting equity. The firm will not declare bankruptcy
as long as the loan commitment is still available. While the firm is above the cash flow
shortage line (C + C l(pt))/q, there is a level of drawn amount, pR, above which the it will
use excess revenue to repay the principle of the loan commitment, and below which it will
use it to pay dividend. Figure 8 illustrates these observations.
25
Figure 8: This figure shows the optimal decisions on the loan commitment. The black solid line is theboundary below which the firm does not generate enough revenue to cover coupon payments. The firm drawsdown the loan commitment below this line, and only dilutes equity or declare bankruptcy when the creditlimit is reached. Above this line, there is a critical level of amount the loan commitment drawn, below whichthe firm uses excess revenue to pay dividend and above which the firm uses it to repay loan commitment.(V = 100, σ = 0.2, r = 0.05, q = 0.03, α = 0.3, τ = 0, γ = 0.05, C = 3.17, P l = 25, Cl = 1.285 (5.14%))
5.2 Impacts of Loan Commitment
In this section, we first discuss the impact of the loan commitment on existing debt and
firm values. The left panel of Figure 9 shows how equity value changes for various sizes
of the loan commitment. We find that the presence of the loan commitment increases the
equity value. This result is intuitive since the loan commitment helps the firm avoid costly
equity dilution. However, we observe that the size of the loan commitment has very small
impact on the change in equity value, just as we previously observed in Figure 7.
We find more interesting results for debt values. While the shareholders benefit from the
loan commitment regardless of the size of the existing debt, we find that the existing debt
holders only benefit if its size, as characterized by the coupon C, is larger than a certain
level. When C is small, the debt holders are better off without the loan commitment. The
26
right panel of Figure 9 shows that with a loan commitment of size $25, the debt holders are
better of if the coupon is greater than $7, whereas for a loan commitment of size $50, the
debt holders are always better off. The loan commitment has both positive and negative
impacts on the debt holders. First, since it is more senior than the existing debt, in the
case of bankruptcy, the debt holders receive less money when the firm is liquidated. This,
and the cost of the loan commitment itself, reduces the existing debt value. Second, it can
help the firm avoid or postpone bankruptcy and hence increase the existing debt value.
Therefore, it is not surprising that the debt holders are worse off when C is small. This is
because the firm is not likely to declare bankruptcy in the future and the costs of having
the loan commitment outweigh the benefits. However, at higher leverage, the probability of
declaring bankruptcy is more significant and the benefits of the loan commitment dominates
the costs.
Figure 9: The left panel plots equity value vs. coupon rate for different sizes of loan commitment. Loancommitment always increases the equity value but its size has little impact on the difference. The rightpanel plots debt value vs. coupon rate. Debt holders do not always benefit from the loan commitment. Highleveraged firms benefit more than low leveraged firms. The debt capacity of the firm increases significantlyin presence of loan commitment, and is increasing in its size, P l. (V = 100, σ = 0.2, r = 0.05, q = 0.03, α =0.3, τ = 0.15, γ = 0.05)
Notice also in the absence of loan commitments, the debt value is constant when the coupon
is above a certain level ($9 in our example). At this level of coupon, the optimal bankruptcy
boundary is above the current asset value, and thus the firm declares bankruptcy imme-
diately, leaving the debt holder with only the liquidated asset value less bankruptcy cost,
27
(1 − α)V . As we discussed previously, it is never optimal to declare bankruptcy when the
firm can still draw down the loan commitment. For this reason, we observe that the debt
value does not behave the same way when a loan commitment is present. Furthermore, the
benefit of a loan commitment increases with its size, because it can prolong the life of the
firm further, and the debt holder gets to collect more coupon payments. In addition, the
debt capacity of the firm significantly increases in the presence of a loan commitment for
the same reason.
Figure 10 shows how the loan commitment increases the total firm value. What is more
interesting here is how it impacts the optimal capital structure. We observe that the optimal
leverage of the firm increases with the size of the loan commitment. This can be explained
by the fact that the loan commitment allows the firm take on more debt to utilize the tax
benefits, while providing it with protection from bankruptcy.
Figure 10: This figure plots total firm value vs. coupon rate for different sizes of loan commitment. Thefirm value increases with the size of the loan commitment. This figure also illustrates how loan commitmentimpacts the optimal capital structure of the firm. The firm with loan commitment will issue debt with muchhigher coupon rate at optimal capital structure. (V = 100, σ = 0.2, r = 0.05, q = 0.03, α = 0.3, τ = 0.15, γ =0.05)
The results from this section leads to the following important observation.
Even in the absence of equity dilution and liquidation costs, it is generally beneficial for the
firm to enter into a loan commitment contract.
28
Note that the firm is able derive benefits by exercising the loan commitment in bad states to
meet any liquidity needs. The critical point is that the firm can optimally stagger the draw
on a loan commitment and only pay (higher) interest on the drawn portion of it. It is this
property that makes the loan commitment attractive. Figure 11 plots the increase in equity
and total firm value from the loan commitment. The left panel shows that the increase in
equity value is increasing in γ, but decreasing in α. This is because equity holders benefit
mostly from avoiding costly dilution. Hence as liquidation cost increases, the increase in
cost of loan commitment outweighs the increase in benefits. On the other hand, the right
panel shows that the increase total firm value is increasing in both γ and α. This is intuitive
because the loan commitment helps to avoid both equity dilution and liquidation costs.
00.1
0.20.3
0.40.5
00.1
0.20.3
0.40.5
0
5
10
15
Dilution Cost, !
Benefits of Loan Commitment on Equity Value
Liquidation Cost, "
Incr
ease
in E
quity
Val
ue
00.1
0.20.3
0.40.5
00.1
0.20.3
0.40.5
0
5
10
15
20
Dilution Cost, !
Benefits of Loan Commitment on Firm Value
Liquidation Cost, "
Incr
ease
in F
irm V
alue
Figure 11: The left panel plots the increase in equity values from loan commitment for different level ofdilution and liquidation costs. The increase in equity value is increasing in γ, but decreasing in α. Theright panel plots the increase in total firm value. The increase in the firm value is increasing in γ and α. Inboth plots, we fix the coupon rate of original debt to $5.27, which is the optimal level for α = 0 and γ = 0.(V = 100, σ = 0.2, r = 0.05, q = 0.03, C = 5.27, α = 0.3, τ = 0.15, γ = 0.05)
As stated above, entering into a loan commitment is generally beneficial for the firm since it
postpones costly equity dilutions and liquidations. Figure 12 shows the probability density
of the time of default with and without the loan commitment. In this example, the expected
time to default increases from 47 to 50 years with a $25 loan commitment. More importantly,
it significantly increases the survival probability in the near future. In this example, the
survival probability is one up to almost ten years, compared with 80% at ten years without
the loan commitment. The probability remains higher for about 50 years. The methodology
for determining the default time distribution is described in Appendix E. The results were
29
obtained by using T = 200, N = 2,400 and K = 50.
Figure 12: The left panel plots the probability density function of the default time with and without theloan commitment. The right panel plots the firm’s survival probability, or the probability that the firm willnot default before a certain time. The expect default time conditioned on ever defaulting increases from 47to 50 years. Moreover, the survival probability over the near future increases significantly in the presence ofloan commitment. (V = 100, σ = 0.2, r = 0.05, q = 0.03, α = 0.3, τ = 0.15, γ = 0.05)
In contrast to cash balance, a loan commitment postpones the default time because the
firm does not need to worry about debt holders taking cash upon liquidation. Moreover,
the firm must first accumulate the cash balance before it can be of any use, while the loan
commitment is available immediately.
6 Conclusion
Our paper characterizes liquidity management strategies and their impact on pricing. We
provide an analytical solution to pricing of the firm’s securities when equity dilution is
costly, and show that costly dilution reduces the firm’s debt capacity and optimal level of
debt. We provide a methodology to investigate the effectiveness of cash balances and loan
commitments as means of managing illiquidity. We find that carrying cash is generally
not efficient even in absence of costs associated with holding cash. This is because a) it
is not immediately available since the firm must accumulate cash first, and b) in doing
so, equity holders forego dividends in good states. On the contrary, we show that loan
30
commitments provide an efficient way to manage illiquidity and can be a better strategy
when equity dilution and liquidation costs are significant. The staggered draw feature
of the loan commitment allows the firm to inject liquidity when needed without foregoing
dividends in the good states. Furthermore, we find that the loan commitment increases both
the debt capacity and the optimal level of debt, reduces default probability, and increases
the expected time to default of the firm.
An interesting extension might be to incorporate strategic debt service in our framework.
This is likely to provide a lower incentive to hold cash balances as strategic debt service is
another way to deal with illiquidity, which imposes greater costs ex-ante to the borrower.
Another extension would be to model the Chapter 11 provisions explicitly in our framework
as Chapter 11 is yet another institution to deal with liquidity problems of companies. We
have not modeled the possibility of carrying cash balances to take advantage of future
investment opportunities. In addition, we have not addressed the fact that companies issue
debt to raise cash for “general corporate purposes.” These are fruitful avenues for further
research.
31
Appendix A
Derivation of equity value from Section 3.1.1
Substituting the general solution (8) into (6), we get:
12σ
2A2Y (Y + 1)V −Y + 12σ
2A3X(X + 1)V −X − (r − q)A1V − (r − q)A2Y V−Y
−(r − q)A3XV−X − r(A0 +A1V +A2V
−Y +A3V−X) + β(qV − C) = 0
(17)
We solve for X and Y by collecting the coefficients of V −X and V −Y in (17) and setting it
to zero:
12σ2A2X(X + 1)− (r − q)A2X − rA2 = 0
⇒ 12σ2X2 +
(12σ2 − (r − q)
)X − r = 0
⇒ X =
(r − q − σ2
2
)+
√(r − q − σ2
2
)2+ 2σ2r
σ2
and Y =
(r − q − σ2
2
)−√(
r − q − σ2
2
)2+ 2σ2r
σ2
A0 and A1 can be solved by collecting the constant and the coefficient of the linear term in
(17) as follows:
−rA0 − βC = 0⇒ A0 = −βCr
(r − q)A1 − rA1 + βq = 0⇒ A1 = β
Similarly, substituting (8) into (7), and solve for A0 and A1 we get:
−rA0 − C = 0⇒ A0 = −Cr
(r − q)A1 − rA1 + q = 0⇒ A1 = 1
32
Since Y ≤ −1, the condition (BC II) in (9) implies that A2 = 0. The remaining coeffi-
cients are determined by solving the system of linear equations imposed by the other three
conditions in (9).
(BC I): β(−Cr + VB
)+A2V
−YB +A3V
−XB = 0
(CC): β(−Cr + C
q
)+A2
(Cq
)−Y+A3
(Cq
)−X= −C
r + Cq + A3
(Cq
)(SP): β −A2Y
(Cq
)−Y−1−A3X
(Cq
)−X−1= 1− A3X
(Cq
)−X−1
And we get the expression for A2, A3 and A3 as given in (10).
Derivation of debt value from Section 3.1.2
Substituting the general solution (12) into (11), we get:
12σ2B2X(X+1)V −X+(r−q)B1V −(r−q)B2XV
−X−r(B0 +B1V +B2V−X)+C = 0 (18)
B0 and B1 can be solved by collecting the constant and the coefficient of the linear term in
(18) as follows:
−rB0 + C = 0⇒ B0 =C
r
(r − q)B1 − rB1 = 0⇒ B1 = 0
The condition (BC I) in (13) implies that:
C
r+B2V
−XB = (1− α)VB
⇒ B2 =(
(1− α)VB −C
r
)1
V −XB
Hence the debt value is as given in (14).
33
Appendix B
Proof of Proposition 1
Define the notation:
xt = cash balance
δt = revenue generated during time interval (t− h, t]
ζt = coupon payment during time interval (t− h, t]
d = dividend payout
Et(xt, d) = Equity value at time t, with cash balance x, and dividend payout d
rx = interest earned on cash balance
p = probability of an up move
Proof. We will establish by induction that Et(x, d) in increasing in d for all t and x, and
hence it is optimal to pay as much dividend as possible, leaving no cash inside the firm. For
convenience, we set τ = 0 without loss of generality.
Let x+ be the value of cash balance at the next time period, i.e.
x+ = erxh (δt + x− ζ − d) (19)
At maturity,
ET (x) = (x+ δT + VT − ζ − P )+ (20)
Let Et[Et+h(x)] = pEut+h(x) + (1 − p)Edt+h(x), where Eut+h(x) and Edt+h(x) are the equity
values at the up and down node at time t+ h respectively.
34
Then, for non-bankruptcy nodes at time T − h,
ET−h(x, d) = e−rhET−h[ET (x+)
]+ d
= e−rhET−h[(x+ + VT + δT − ζ − P )+
]+ d
Case I: For a range of d such that both EuT (x+) and EdT (x+) are not bankrupt,
ET−h(x, d) = e−rhET−h[x+ + VT + δT − ζ − P
]+ d
= e−rh(x+ + ET−h[VT + δT ]− ζ − P ) + d
⇒ ∂ET−h∂d
= e−rh∂x+
∂d+ 1
= 1− e−(r−rx)h ≥ 0 for rx ≤ r (21)
Case II: EdT (x+) is bankrupt, but EuT (x+) is not,
ET−h(x, d) = e−rhp(x+ + V uT + δuT − ζ − P ) + d
⇒ ∂ET−h∂d
= 1− pe−(r−rx)h ≥ 0 for rx ≤ r
Case III: Both EuT (x+) and EdT (x+) are bankrupt
ET−h(x, d) = d
⇒ ∂ET−h∂d
= 1
Since, ET−h is increasing in d for all cases, the optimal dividend at time T − h, d∗T−h is
given by
d∗T−h = (x+ δT−h − ζ)+ (22)
When the firm does not have enough revenue and cash to cover the coupon payments, it
has to dilution equity, which is equivalent to paying a negative dividend. So to cover the
35
case where dilution occurs, we simply write the optimal dividend as
d∗T−h = x+ δT−h − ζ (23)
Let E∗T−h(x) = ET−h(x, d∗T−h). Then for the three cases above, we have
E∗T−h(x) =
e−rh(ET−h[VT + δT ]− ζ − P ) + x+ δT−h − ζ
e−rhp(V uT + δuT − ζ − P ) + x+ δT−h − ζ
x+ δT−h − ζ
⇒∂E∗T−h∂x
= 1
Assume that for non-bankruptcy nodes at time t+ h,∂E∗
t+h
∂x = 1. Then at time t,
Case I: For a range of d such that both Eut+h(x+) and Edt+h(x+) are not bankrupt,
Et(x, d) = e−rhEt[E∗t+h(x+)
]+ d
⇒ ∂E∗t∂d
= e−rh(∂E∗t+h∂x+
∂x+
∂d
)+ 1
= 1− e−(r−rx)h ≥ 0 for rx ≤ r
Case II: Edt+h(x+) is bankrupt, but Eut+h(x+) is not,
Et(x, d) = e−rhpEu∗t+h(x+) + d
⇒ ∂Et∂d
= pe−rh(∂E∗t+h∂x+
∂x+
∂d
)+ 1
= 1− pe−(r−rx)h ≥ 0 for rx ≤ r
Case III: Both nodes Eut+h(x+) and Edt+h(x+) are bankrupt
Et(x, d) = d
⇒ ∂Et∂d
= 1
36
Hence Et(x, d) is increasing in d, and d∗t = x + δt − ζ is the optimal dividend payout. We
can write the optimal equity value for the three cases as,
E∗t (x) =
e−rhEt[E∗t+h(0)] + x+ δt − ζ
e−rpEu∗t+h(0) + x+ δt − ζ
x+ δt − ζ
⇒ ∂E∗t∂x
= 1
which completes the proof.
Appendix C
Binomial Lattice for Cash Balance
We extend the binomial lattice method of Broadie and Kaya (2007) to price corporate debt
in the presence of cash balances. Following ?, we divide time into N increments of length
h = T/N , then we write the evolution of the firm asset value Vt as follows:
Vt+h =
uVt with risk-neutral probability p
dVt with risk-neutral probability 1− p(24)
Let V ut = uVt and V d
t = dVt. By imposing that u = 1/d, we have:
u = eσ√h (25)
d = e−σ√h (26)
a = e(r−q)h (27)
p =a− du− d
(28)
37
In this setting, we approximate revenue during time interval h by:
∫ t+h
tqVtdt ≈ Vt(eqh − 1) =: δt
Similarly, we approximate the coupon payment by:
∫ t+h
tcPdt ≈ P (ech − 1) =: ζ (29)
Incorporating the tax benefit of rate τ , the effective coupon payment is (1− τ)ζ.
Each node in the lattice is a vector of dimension M , where each element in the vector
corresponds to a different level of cash balance, x. The values of x are linearly spaced from
0 to xmax.5
We distinguish the cash before and cash after revenue and dividend by x and x+, respec-
tively.
Let Et(Vt, xt) be the equity value at time t when the firm’s asset value is Vt, and its
cash balance level is xt. Similarly, Dt(Vt, xt) is the corresponding debt value. We also let
Et[ft+h(Vt+h, xt+h)] be the expected value of the security f(·) at the next time period t+h,
given the information at time t, where f(·) can represent the equity value, E(·), or the debt
value, D(·), namely,
Et[ft+h(Vt+h, xt+h)] = pft+h(V ut+h, xt+h) + (1− p)ft+h(V d
t+h, xt+h).
Note that xt+h is deterministic because it only depends on the decision at time t. Since we
only know the values of ft+h(V, x) for a discrete set of values of x, we use linear interpolation
to estimate ft+h(V ut+h, xt+h) and ft+h(V d
t+h, xt+h).
Figure 13 illustrates what each node in the lattice represents.
At maturity, the firm returns everything to shareholders after meeting its debt obligations.5In our routine, we pick xmax arbitrarily and check if the optimal cash balance, x∗, is attained at xmax.
If it does, then we increase xmax and repeat the routine.
38
! "0,tt Vf . . .
! "max, xVf tt
! "0,uhtht Vf ##
.
.
. ! "max, xVf u
htht ##
! "0,dhtht Vf ##
.
.
. ! "max, xVf d
htht ##
p
1 – p
Figure 13: This figure illustrates the vector in each node of the lattice. ft(·) represents the security valueat each element of the vector, where ft can represent the equity value, Et, or the debt value, Dt.
If VT + xT + δT ≥ (1− τ)ζ + P :
ET (VT , xT ) = VT + xT + δT − (1− τ)ζ − P
DT (VT , xT ) = ζ + P
If (VT , xT ) is such that the firm does not have enough value to meet its debt obligation, it
declares bankruptcy and liquidates its assets.
If VT + xT + δT < (1− τ)ζ + P :
ET (VT , xT ) = 0
DT (VT , xT ) = (1− α)(VT + xT + δT )
At other times t, if (Vt, xt) is such that the firm has enough cash from revenue and cash
balance to meet its debt obligations, then it has an option to retain some excess revenue in
the cash balance, or pay it out to shareholders as dividend.
39
1. If xt + δt ≥ (1− τ)ζ:
Choose x+t ∈ [0, xt + δt − (1− τ)ζ]
xt+h = erxhx+t
Et(Vt, xt) = e−rhEt [Et+h(Vt+h, xt+h)] + xt + δt − (1− τ)ζ − x+t
Dt(Vt, xt) = e−rhEt [Dt+h(Vt+h, xt+h)] + ζ
x+t is chosen such that the objective value (equity or firm value) is maximized at time t.
The optimization is done by performing a search over K discrete points of feasible values
of x+t as given above.
If (Vt, xt) is such that the firm does not have enough from revenue and cash balance, then
it has to raise money by diluting equity.
2. If xt + δt < (1− τ)ζ:
If (Vt, xt) is such that the firm has enough equity to cover the shortfall:
2A. If (1− γ)e−rhEt[Et+h(Vt+h, 0)] ≥ (1− τ)ζ − xt − δt:
Et(Vt, xt) = e−rhEt [Et+h(Vt+h, 0)] + (xt + δt − (1− τ)ζ)/(1− γ)
Dt(Vt, xt) = e−rhEt [Dt+h(Vt+h, 0)] + ζ
If (Vt, xt) is such that the firm does not have enough equity value to cover the shortfall,
then it declares bankruptcy:
2B. If (1− τ)ζ − xt − δt > (1− γ)e−rhEt[Et+h(Vt+h, 0)]
Et(Vt, xt) = 0
Dt(Vt, xt) = (1− α)(Vt + xt + δt)
In the infinite maturity case, we use an arbitrarily large maturity T and modify the terminal
values of states (VT , xT ) such that the firm is not bankrupt as follows:
40
If VT + xT + δT ≥ (1− τ)ζ + P :
ET (VT , xT ) = VT + xT − (1− τ)ζ/(1− r)
DT (VT , xT ) = ζ/(1− r)
We use the infinite horizon implementation with T = 200, N = 2,400 and M = 40 for the
results in Figures 3 – 6.
Appendix D
Binomial Lattice for Loan Commitment
The basic setup of the lattice is as described in Appendix C. Each node in the lattice is a
vector of dimension M , where each element in the vector corresponds to the different level
of outstanding loan commitment, p. The values of p are linearly spaced from 0 to P l.
Let λt be the amount the loan commitment drawn at time t. In addition to Et(pt) and
Dt(pt), we let Lt(pt) be the loan commitment value to the bank at time t, with the drawn
amount of pt, and Λt(pt) be its value to the firm (i.e., future cash flow from the loan
commitment to the firm). Again, we let Et[ft+h(Vt+h, xt+h)] be the expected value of the
security f(·) at the next time period t+ h, given the information at time t, where f(·) can
represent L(·) or Λ(·), in addition to E(·) and D(·).
At maturity, the firm returns everything to shareholders after meeting all of its debt obli-
gations.
41
If VT + δT ≥ (1− τ)(ζ + ζ l(pT )) + P + pT :
ET (VT , pT ) = VT + δT − (1− τ)(ζ + ζ l(pT ))− (P + pT )
LT (VT , pT ) = ζ l(pT ) + pT
DT (VT , pT ) = ζ + P
ΛT (VT , pT ) = 0
If (VT , pT ) is such that the firm does not have enough value to meet its debt obligation,
then it declares bankruptcy and the liquidation cost is shared among the debt holders as
described in Section 5.
If VT + δT < (1− τ)(ζ + ζ l(pT )) + P + pT :
ET (VT , pT ) = 0
LT (VT , pT ) = min{VT + δT , ζ
l(pT ) + pT
}DT (VT , pT ) = VT + δT − LT (VT , pT )
LT (VT , pT ) = LT (VT , pT )− LT (VT , pT )LT (VT , pT ) + DT (VT , pT )
α(VT + δT )
DT (VT , pT ) = DT (VT , pT )− DT (VT , pT )LT (VT , pT ) + DT (VT , pT )
α(VT + δT )
ΛT (VT , pT ) = 0
At other times t, if (VT , pT ) is such that the firm generates enough revenue to meet its debt
obligations, then it has an option to repay some of the loan commitment by using excess
cash from revenue or equity dilution, or both. Any remaining cash goes to shareholders as
dividend.
1. If δt ≥ (1− τ)(ζ + ζ l(pt)):
Choose λt ∈ [−pt, 0]
pt+h = pt + λt
Λt(Vt, pt) = e−rhEt [Λt+h(Vt+h, pt+h)]
42
If (Vt, pt) is such that the firm has enough equity value to repay the chosen amount of
loan commitment, λt:
1A. If (1− γ)Et [Et+h(Vt+h, pt+h)] ≥ (1− τ)(ζ + ζ l(pt))− δt − λt:
Et(Vt, pt) = e−rhEt [Et+h(Vt+h, pt+h)] + min{δt − (1− τ)(ζ + ζ l(pt)) + λt,(δt − (1− τ)(ζ + ζ l(pt)) + λt
)/(1− γ)}
Lt(Vt, pt) = e−rhEt [Lt+h(Vt+h, pt+h)] + ζ l(pt)− λt
Dt(Vt, pt) = e−rhEt [Dt+h(Vt+h, pt+h)] + ζ
If (Vt, pt) is such that the firm does not have enough equity value, it declares bankruptcy:
1B. If (1− τ)(ζh+ ζ l(pt))− δt − λt > (1− γ)e−rhEt[Et+h(Vt+h, pt+h)]:
Et(Vt, pt) = 0
Lt(Vt, pt) = min{Vt + δt, ζ
l(pt) + pt
}Dt(Vt, pt) = Vt + δt − Lt(Vt, pt)
Lt(Vt, pt) = Lt(Vt, pt)−Lt(Vt, pt)
Lt(Vt, pt) + Dt(Vt, pt)α(Vt + δt)
Dt(Vt, pt) = Dt(Vt, pt)−Dt(Vt, pt)
Lt(Vt, pt) + Dt(Vt, pt)α(Vt + δt)
If (Vt, pt) is such that the firm does not generate enough revenue, then it has to raise money
by either drawing down the loan commitment or diluting equity. The maximum it can draw
from the loan commitment is limited by the credit limit P l, and the maximum it can raise
from equity dilution is limited by its equity value.
2. If δt < (1− τ)(ζ + ζ l(pt)):
Choose λt ∈ [0,min{(1− τ)(ζ + ζ l(pt))− δt, P lmax − pt]
pt+h = pt + λt
Λt(Vt, pt) = e−rhEt [Λt+h(Vt+h, pt+h)] + λt
43
2A. If (Vt, pt) is such that λt alone is enough to cover the shortfall, i.e., λt = (1 −
τ)(ζ + ζ l(pt))− δt, then:
Et(pt) = e−rhEt [Et+h(pt+h)]
Lt(pt) = e−rhEt [Lt+h(pt+h)] + ζ l(pt)
Dt(pt) = e−rhEt [Dt+h(pt+h)] + ζ
If (Vt, pt) is such that λt is not enough to cover the shortfall, i.e., λt < (1 − τ)(ζ +
ζ l(pt))− δt, and the firm has enough equity to cover the difference:
2B. If (1− γ)e−rhEt[Et+h(Vt+h, pt+h)] ≥ (1− τ)(ζ + ζ l(pt))− δt − λt:
Et(Vt, pt) = e−rhEt [Et+h(Vt+h, pt+h)] + (δt − (1− τ)(ζ + ζ l(pt)) + λt)/(1− γ)
Lt(Vt, pt) = e−rhEt [Lt+h(Vt+h, pt+h)] + ζ l(pt)
Dt(Vt, pt) = e−rhEt [Dt+h(Vt+h, pt+h)] + ζ
If at the chosen level of λt, (Vt, pt) is such that the firm does not have enough equity
value to cover the shortfall, then it declares bankruptcy:
2C. If (1− τ)(ζ + ζ l(pt))− δt − λt > (1− γ)e−rhEt[Et+h(Vt+h, pt+h)]
Et(Vt, pt) = 0
Lt(Vt, pt) = min{Vt + δt, ζ
l(pt) + pt
}Dt(Vt, pt) = Vt + δt − Lt(pt)
Lt(Vt, pt) = Lt(Vt, pt)−Lt(Vt, pt)
Lt(Vt, pt) + Dt(Vt, pt)α(Vt + δt)
Dt(Vt, pt) = Dt(Vt, pt)−Dt(Vt, pt)
Lt(Vt, pt) + Dt(Vt, pt)α(Vt + δt)
In the infinite maturity case, we use an arbitrarily large maturity T and modify the terminal
values of states (VT , xT ) such that the firm is not bankrupt as follows:
44
If VT + δT ≥ (1− τ)(ζ + ζ l(pT )) + P + pT :
ET (VT , pT ) = VT + δT − (1− τ)(ζ + ζ l(pT ))/(1− r)
LT (VT , pT ) = ζ l(pT )/(1− r)
DT (VT , pT ) = ζ/(1− r)
We use the infinite horizon implementation with T = 200, N = 2,400 and M = 40 for the
results in Figures 7 – 12.
Appendix E
Estimating Default Distribution in Binomial Lattice
In this section, we describe a method for estimating the distribution of default time in the bi-
nomial lattice. For simplicity, we assume that there is no cash balance or loan commitment,
but the method can be easily applied to both cases.
We discretize the probability density of default time into K buckets of time intervals of
length ∆ = T/K. The kth bucket corresponds to the time interval (tk, tk+1], for k =
0, 1, ...,K− 1, where tk = k∆. K can be any positive integer no greater than the number of
discrete time steps, N , in the lattice. Let Πt(Vt, k) be the time t probability of defaulting
during the kth interval when the firm’s asset value is Vt.
At maturity, if VT is such that the firm is not bankrupt, then the default probability is zero
for all intervals k.
ΠT (VT , k) = 0 for all k
45
Otherwise, if VT is such that the firm is bankrupt, then the probability of defaulting at
t = T is one, and is zero everywhere else:
ΠT (VT , k) =
0 for k 6= K − 1
1 for k = K − 1
because T is in the (K − 1)st interval (the last interval).
At other times t, if Vt is such that the firm is not bankrupt, then the time t distribution of
default time is the expected value of time t+ h distribution.
Πt(Vt, k) = Et[Πt+h(Vt+h, k)] for all k
Otherwise, if Vt is such that the firm is bankrupt, then probability of defaulting during the
interval where t belongs in is one, and is zero everywhere else.
Πt(Vt, k) =
0 for k such that t /∈ (tk, tk+1]
1 for k such that t ∈ (tk, tk+1]
Figure 14 provides an example of the default time distribution in a binomial lattice with
K = N = 4 and p = 0.5.
We use T = 200, N = 2,400 and K = 50 for the results in Figure 12 and 16.
Appendix F
Effects of Cash Balance on Default Time
As discussed in Section 4.2, cash balance does not have a significant impact on default
time when the firm is maximizing the equity value. This is because the firm depletes the
cash balance so that shareholders can receive more dividends prior to bankruptcy. The
impact of cash balance on default time is significantly different under total firm value
maximization. Figure 15 plots the optimal level of cash balance as a function of asset
46
000
0 00 00
0 0 00 0 00 0
Bankrupt? 0 0 0 0!t(Vt, 0) 0 1/4 0 0!t(Vt, 1) 1/2 0!t(Vt, 2) 0 1 0 1!t(Vt, 3) 1/8 0 1/2 0
1 00 1 00 0 1
01 10 0
001
Figure 14: This figure shows the default time distribution at each node of the binomial lattice for K =N = 4 and p = 0.5. Each box represents a node in the lattice. The first (bold) number is 1 if the firm isbankrupt at that node, or is 0 otherwise. The remaining four elements are the probabilities of defaulting attime 0, 1, ... , 4, respectively.
value. Unlike the equity maximization case, the optimal cash level increases as the asset
value decreases, regardless of how close it is to bankruptcy. Consequently, as the firm
heads towards bankruptcy, it will not empty out the cash balance and will be able to
remain solvent longer. This has a big impact on the probability of default. In our example,
the probability of default reduces from 71.3% to 40.0%. However, since there is a huge
reduction in probability of default, the expected default time, conditioned on ever defaulting,
reduces from 44.6 to 19.6. Figure 16 shows the distribution of default time and the survival
probability. We find that the probability of defaulting within ten years is practically the
same in both cases. The probabilities start to deviate at later times. The reason for this
is because in the next few years, the firm is not likely to be able to accumulate significant
amount of cash to make a difference. If the firm survives the initial period, then the survival
probability is significantly different.
47
Figure 15: This figure plots the optimal level of cash balance, x∗, vs. asset value, V , under firm valuemaximization. Unlike equity value maximization, x∗ is decreasing in V for all values of V . (V = 100, C =3.36, σ = 0.2, r = 0.05, q = 0.03, α = 0.3, τ = 0.15)
Figure 16: The left panel plots the probability density function of the default time with and without cashbalance under firm value maximization. The distribution is significantly different because the firm keepscash inside even as it approaches bankruptcy. The right panel plots the survival probability. The survivalprobability is almost the same for t ≤ 10, but starts to deviate (upward) significantly for larger values of t.(V = 100, C = 3.36, σ = 0.2, r = 0.05, q = 0.03, α = 0.3, τ = 0.15)
References
V. Acharya, J.-Z. Huang, M. Subrahmanyam, and R. K. Sundaram. When does Strategic
Debt-Service Matter? Economic Theory, 29(2):363–378, 2006.
48
S. Agarwal, S. Chomsisengphet, and J. C. Driscoll. Loan Commitments and Private Firms.
Finance and Economics Discussion Series 2004-27, Board of Governors of the Federal
Reserve System (U.S.), 2004.
P. Aghion, O. Hart, and J. Moore. The Economics of Bankruptcy Reform. Journal of Law,
Economics and Organization, 8:523–546, 1992.
R. W. Anderson and S. M. Sundaresan. Design and Valuation of Debt Contracts. Review
of Financial Studies, 9(1):37–68, 1996.
S. Asmussen and M. I. Taksar. Controlled Diffusion Models for Optimal Dividend Pay-out.
Insurance: Mathematics and Economics, 20(1):1–15, 1997.
L. A. Bebchuk. A New Approach to Corporate Reorganization. Havard Law Review, 101:
775–804, 1988.
L. A. Bebchuk and J. M. Fried. The Uneasy Case for the Priority of Secured Claims in
Bankruptcy. Yale Law Journal, 105:857–934, 1996.
E. Berkovitch and S. I. Greenbaum. The Loan Commitment as an Optimal Financing
Contract. Journal of Financial and Quantitative Analysis, 26(1):83–95, 1991.
F. Black and J. C. Cox. Valuing Corporate Securities: Some Effects of Bond Indenture
Provisions. Journal of Finance, 31(2):351–367, 1976.
B. Branch. The Costs of Bankruptcy: A Review. International Review of Financial Anal-
ysis, 11(1):39–57, 2002.
A. Bris, I. Welch, and N. Zhu. The Costs of Bankruptcy: Chapter 7 Liquidation versus
Chapter 11 Reorganization. Journal of Finance, 61(3):1253–1303, 2006.
M. Broadie and O. Kaya. A Binomial Lattice Method for Pricing. Corporate Debt and
Modeling Chapter 11 Proceedings. Journal of Financial and Quantitative Analysis, 42
(2), 2007.
M. Broadie, M. Chernov, and S. M. Sundaresan. Optimal Debt and Equity Values in the
Presence of Chapter 7 and Chapter 11. Journal of Finance, 62, 2007.
49
A. Cadenillas, S. Sarkar, and F. Zapatero. Optimal Dividend Policy with Mean-Reverting
Cash Reservoir. Mathematical Finance, 17(1):81–109, 2007.
D. Duffie. An Extension of the Black-Scholes Model of Security Valuation. Journal of
Economic Theory, 46(1):194–204, 1988.
H. Fan and S. M. Sundaresan. Debt Valuation, Renegotiations, and Optimal Dividend
Policy. The Review of Financial Studies, 13(4):1057–1099, 2000.
O. Hart. Different Approaches to Bankruptcy. Harvard Institute of Economic Researh
Paper 1903, Harvard Institute of Economic Research, 2000.
O. Hart, R. La Porta Drago, F. Lopez-de Silane, and J. Moore. A New Bankruptcy Proce-
dure that Uses Multiple Auctions. European Economic Review, 41(3):461–473, 1997.
G. D. Hawkins. An Analysis of Revolving Credit Agreements. Journal of Financial Eco-
nomics, 10(1):59–81, 1982.
T. S. Y. Ho and A. Saunders. Fixed Rate Loan Commitments, Take-Down Risk, and the
Dynamics of Hedging with Futures. Journal of Financial and Quantitative Analysis, 18
(4):499–516, 1983.
C. M. James. Credit Market Conditions and the Use of Bank Lines of Credit. Federal
Reserve Bank of San Francisco Economic Letter, 27, 2009.
M. Jeanblanc-Picque and A. N. Shiryaev. Optimization of the Flow of Dividends. Russian
Mathematical Surveys, 50:257–277, 1995.
C.-S. Kim, D. C. Mauer, and A. E. Sherman. The Determinants of Corporate Liquidity:
Theory and Evidence. Journal of Financial and Quantitative Analysis, 33(3):335–359,
1998.
I. Lee, S. Lochhead, J. Ritter, and Q. Zhao. The Costs of Raising Capital. Journal of
Financial Research, 19(1):59–74, 1996.
H. E. Leland. Corporate Debt Value, Bond Covenants, and Optimal Capital Structure.
Journal of Finance, 49(4):1213–1252, 1994.
50
V. Maksimovic. Product Market Imperfections and Loan Commitments. Journal of Finance,
45(5):1641–53, 1990.
J. S. Martin and A. M. Santomero. Investment Opportunities and Corporate Demand for
Lines of Credit. Journal of Banking & Finance, 21(10):1331–1350, 1997.
A. Melnik and S. Plaut. Loan Commitment Contracts, Terms of Lending, and Credit
Allocation. Journal of Finance, 41(2):425–35, 1986.
R. C. Merton. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.
Journal of Finance, 29(2):449–470, 1974.
M. H. Miller and F. Modigliani. Dividend Policy, Growth, and the Valuation of Shares.
Journal of Business, 34(4):411–433, 1961.
S. C. Myers. Determinants of Corporate Borrowing. Journal of Financial Economics, 5(2):
147–175, 1977.
C. M. Pedersen. On Corporate Debt and Credit Risk. PhD thesis, Columbia University,
2004.
R. Radner and L. Shepp. Risk vs. Profit Potential: A Model for Corporate Strategy. Journal
of Economic Dynamics and Control, 20(8):1373–1393, 1996.
R. L. Shockley. Bank Loan Commitments and Corporate Leverage. Journal of Financial
Intermediation, 4(3):272–301, 1995.
R. L. Shockley and A. V. Thakor. Bank Loan Commitment Contracts: Data, Theory, and
Tests. Journal of Money, Credit and Banking, 29(4):517–34, 1997.
C. M. Snyder. Loan Commitments and the Debt Overhang Problem. Journal of Financial
and Quantitative Analysis, 33(1):87–116, 1998.
A. Sufi. Information Asymmetry and Financing Arrangements: Evidence from Syndicated
Loans. Journal of Finance, 62(2), 2007a.
A. Sufi. Bank Lines of Credit in Corporate Finance: An Empirical Analysis. Forthcoming
in Review of Financial Studies, 2007b.
51
M. I. Taksar. Optimal Risk and Dividend Distribution Controls Models for an Insurance
Company. Mathematical Methods of Operations Research, 51:1–42, 2000.
52